Complex multiple feedback filter

ABSTRACT

A complex filter includes an I channel having a first I channel output and a second I channel output and a Q channel having a first Q channel output and a second Q channel output. The second I channel output is input to the Q channel through a first passive network and wherein the second Q channel output is input to the I channel through a second passive network.

FIELD OF THE INVENTION

The present invention relates generally to a filter. An image rejectionfilter with two poles per op amp is disclosed.

BACKGROUND OF THE INVENTION

There is a need for better image rejection filters to be used inwireless systems. In general, an image rejection filter is implementedas a bandpass filter that passes only positive frequencies. Since themagnitude of the transfer function is not symmetric about DC, thetransfer function is complex. FIG. 1 is a diagram illustrating a lowpass to bandpass filter transformation. Since the filter shown requirescomplex signals, it needs to be implemented as a filter with two realinputs. FIG. 2 is a block diagram illustrating an implementation of sucha complex filter. Input 202 is referred to as the in phase or I inputand input 204 is referred to as the quadrature or Q input. As shown inFIG. 2, the I channel and the Q channels are linked. An output QO of theQ channel 214 is fed into a second input I2 to the I channel 206.Likewise, an output IO of the I channel 212 is fed into a second inputQ2 to the Q channel 208.

Ideally, to implement a complex filter as shown in FIG. 2 it would beuseful to have a filter that has a minimum number of op amps or otheractive components. “CMOS Mixers and Polyphase Filters for Large ImageRejection” by Farbod Behbahani, Yoji Kishigami, John Leete, and Asad A.Abidi, published in the IEEE Journal of Solid-State Circuits, Vol. 36,No. 6, June 2001, page 873 and 878 describes a polyphase multiple polefilter that includes several RC stages. The design is useful, but it isdifficult to implement desired types of bandpass filters and themultiple stages attenuate the signal so that amplification of the outputis required. “An Image-Rejecting Mixer and Vector Filter with 55-dBImage Rejection over Process, Temperature, and Transistor Mismatch,”Thomas Homak, Knud L. Knudsen, Anderew Z. Grezegorek, Ken A. Nishimura,and William J. McFarland, IEEE Journal of Solid-State Circuits, Vol. 36,No. 1, January 2001, page 23 and 26 describes an image rejection filterthat integrates op amps into the filter design so that amplification isbuilt into the filter. This design also has the advantage that desiredtypes of bandpass filter responses can be implemented. However, theillustrated design only provides one pole per op amp. A complex multiplepole filter therefore would require an op amp for each pole in eachchannel. It would be desirable if a filter could be designed that couldbe used in an image rejection system and would use fewer op amps perpole. Such a filter could use less power and take up less space on achip than currently available designs.

SUMMARY OF THE INVENTION

A complex multiple feedback filter is disclosed. A multiple feedbackfilter that realizes two poles and a single op amp is implemented in theI and Q channels. A first linking network of capacitors and resistorslinks a Q channel output back to an I channel input and a second linkingnetwork of capacitors and resistors links an I channel output back to aQ channel input. In this manner, a complex filter is implemented withtwo poles and only one op amp in the I channel and one op amp in the Qchannel. Higher order complex filters can be implemented by cascadingthe two pole filter design.

It should be appreciated that the present invention can be implementedin numerous ways, including as a process, an apparatus, a system, adevice, a method, or a computer readable medium such as a computerreadable storage medium or a computer network wherein programinstructions are sent over optical or electronic communication links.Several inventive embodiments of the present invention are describedbelow.

In one embodiment, a complex filter includes an I channel having a firstI channel output and a second I channel output and a Q channel having afirst Q channel output and a second Q channel output. The second Ichannel output is input to the Q channel through a first passive networkand wherein the second Q channel output is input to the I channelthrough a second passive network.

These and other features and advantages of the present invention will bepresented in more detail in the following detailed description and theaccompanying figures which illustrate by way of example the principlesof the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be readily understood by the followingdetailed description in conjunction with the accompanying drawings,wherein like reference numerals designate like structural elements, andin which:

FIG. 1 is a diagram illustrating a low pass to bandpass filtertransformation.

FIG. 2 is a block diagram illustrating an implementation of such acomplex filter.

FIG. 3 is a block diagram illustrating a multiple feedback filter 300with a second order transfer function having 2 poles and a single opamp.

FIG. 4 is a block diagram illustrating a filter.

FIG. 5 is a block diagram illustrating a filter.

FIG. 6 is a diagram illustrating a desired filter configuration.

FIG. 7 is a diagram illustrating a passive network.

FIG. 8 is a diagram illustrating a network.

FIG. 9 is a diagram illustrating a network.

FIG. 10 is a block diagram illustrating two second order filters eachimplemented using a single op amp.

DETAILED DESCRIPTION

A detailed description of a preferred embodiment of the invention isprovided below. While the invention is described in conjunction withthat preferred embodiment, it should be understood that the invention isnot limited to any one embodiment. On the contrary, the scope of theinvention is limited only by the appended claims and the inventionencompasses numerous alternatives, modifications and equivalents. Forthe purpose of example, numerous specific details are set forth in thefollowing description in order to provide a thorough understanding ofthe present invention. The present invention may be practiced accordingto the claims without some or all of these specific details. For thepurpose of clarity, technical material that is known in the technicalfields related to the invention has not been described in detail so thatthe present invention is not unnecessarily obscured.

FIG. 3 is a block diagram illustrating a multiple feedback filter 300with a second order transfer function having 2 poles and a single opamp.

The design equations can be derived as follows:

KCL at node A: $\begin{matrix}{{\frac{\left( {v_{i} - v_{C_{1}}} \right)}{R_{1}} - {{sC}_{1}v_{\; C_{1}}} + \frac{\left( {v_{o} - v_{C_{1}}} \right)}{R_{2}} - \frac{v_{\; C_{1}}}{R_{3}}} = 0} & (1)\end{matrix}$

After rearrangement: $\begin{matrix}{\frac{v_{i}}{R_{1}} = {{- \frac{v_{o}}{R_{1}}} + {v_{\; C_{1}}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)}}} & (2)\end{matrix}$

KCL at Node B: $\begin{matrix}{{\frac{v_{C_{1}}}{R_{3}} + {{sC}_{2}v_{o}}} = {\left. 0\Rightarrow v_{C_{1}} \right. = {{- v_{o}}{sR}_{3}C_{2}}}} & (3)\end{matrix}$

Substituting (3) in (2): $\begin{matrix}{\frac{v_{i}}{R_{1}} = {- {v_{o}\left\lbrack {{{sR}_{3}{C_{2}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)}} + \frac{1}{R_{2}}} \right\rbrack}}} & (4)\end{matrix}$ $\begin{matrix}{{\frac{v_{o}}{v_{i}}(s)} = \frac{H_{0}\omega_{0}^{2}}{s^{2} + {\frac{\omega_{0}}{Q}\; s} + \omega_{0}^{2}}} & (5) \\{= \frac{{- \frac{R_{2}}{R_{1}}}\;\frac{1}{C_{1}C_{2}R_{2}R_{3}}}{s^{2} + {\frac{1}{C_{1}}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}} \right)s} + \frac{1}{C_{1}C_{2}R_{2}R_{3}}}} & (6)\end{matrix}$

By inspection: $\begin{matrix}{H_{0} = {- \frac{R_{2}}{R_{1}}}} & (7)\end{matrix}$ $\begin{matrix}{\omega_{0}^{2} = \frac{1}{C_{1}C_{2}R_{2}R_{3}}} & (8)\end{matrix}$ $\begin{matrix}{\frac{\omega_{0}}{Q} = {\frac{1}{C_{1}}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}} \right)}} & (9)\end{matrix}$

As can be seen from equation (6), multiple feedback filter 300 has asecond order transfer function as desired. To implement an arrangementas shown in FIG. 2, filter 300 is configured to receive two inputs andto have an ouput that drives a filter in another channel. In oneembodiment, one second order filter per channel each having a single opamp is implemented using the configuration derived below.

If the original transfer function is H(jω), the complex band-passtransfer function is H(jω′) where ω′=ω−ω_(c). Since the only frequencydependent components of the filter shown in FIG. 1 are capacitors, thecomplex filter can be obtained from the circuit by transforming eachcapacitor to a composite device as shown in FIG. 4.

The resulting filter would look like FIG. 5 with:i ₁ =−jω _(c) C ₁ v _(A)  (10)i ₂ =−jω _(c) C ₂ v _(o)  (11)

Consider two such filters H_(R) and H_(I) such that H_(R) is excited witthe input signal v_(i), and H_(I) is excited by jv_(i). Denote the nodesvoltages of the corresponding filter with v*_(A) and v*_(o). In otherwords:i ₁=−ω_(c) C ₁ v* _(A)  (12)i ₂=−ω_(c) C ₂ v* _(o)  (13)

The controlled current source in FIG. 5 subtracts current from node Aand depends on the voltage v*_(A) with the coefficient −ω_(c)C₁. Thiscurrent source cannot be implemented with passive components becauseadding passive components to node A changes the transfer function.Furthermore the current depends on the voltage of the correspondingnode, A*, in the complex filter. However, nodes A and A* are not lowimpedance nodes; therefore, they cannot drive impedances withoutbuffers.

In general, it is desired that the driving point for the outputs to theother channel is at the op amp output so that there is low impedance.Also, it is desired that the mixing point at the input is at the op ampinput where there is a virtual ground.

A suitable implementation can be obtained first using v*_(o) as thecontrolling voltage rather than v*_(A): $\begin{matrix}{i_{1} = {{- \omega_{c}}C_{1}v_{A}^{*}}} & {\mspace{374mu}(14)} \\{= {{- w_{c}}C_{1}\;\frac{v_{A}}{v_{o}}\; v_{o}^{*}\mspace{14mu}\left( {{{Note}\text{:}\frac{v_{A}}{v_{o}}} = \frac{v_{A}^{*}}{v_{o}^{*}}} \right)}} & {\mspace{374mu}(15)}\end{matrix}$

The transfer function v_(A)/v_(o) can be found by inspection:$\begin{matrix}{\frac{v_{A}}{v_{o}} = {{- R_{3}}{C_{2}\left( {s - {j\;\omega_{c}}} \right)}}} & (16)\end{matrix}$

The next step is to use the op-amp input as the current summingjunction. With this change, passive components between the op-amp inputsand outputs of the complex filters can be used for the cross-couplingelements. FIG. 6 is a diagram illustrating the desired configuration.The controlled current source i₂ can be moved to between op-amp inputand ground without changing the transfer function. Replacing i₁ withi_(x) should produce the same output voltage v_(o). KCL at node A inFIG. 5: $\begin{matrix}{{\frac{v_{A}}{R_{1}} + {sC}_{1} + \frac{v_{A}}{R_{3}} + \frac{v_{A} - v_{o}}{R_{2}} + i_{1}} = 0} & (17) \\{{v_{A}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)} = {\frac{v_{o}}{R_{2}} - i_{1}}} & (18)\end{matrix}$

Using (16): $\begin{matrix}{{{- v_{o}}R_{3}{C_{2}\left( {s - {j\;\omega_{c}}} \right)}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)} = {\frac{v_{o}}{R_{2}} - i_{1}}} & (19)\end{matrix}$

Finally, $\begin{matrix}{v_{o} = {i_{1}\;\frac{1}{R_{3}{C_{2}\left( {s - {j\;\omega_{c}}} \right)}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)\frac{1}{R_{2}}}}} & (20)\end{matrix}$

The output, v_(o) in FIG. 6 can be found similarly. KCL at node A:$\begin{matrix}{{v_{A}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)} = \frac{v_{o}}{R_{2}}} & (21)\end{matrix}$

KCL at node B: $\begin{matrix}{{\frac{v_{A}}{R_{3}} + i_{x} + {v_{o}{C_{2}\left( {s - {j\;\omega_{c}}} \right)}}} = 0} & (22)\end{matrix}$

v _(A) =−i _(x) R ₃ −v _(o) R ₃ C ₂(s−jω _(c))  (23) $\begin{matrix}{{{Substituting}\mspace{14mu}(23)\mspace{14mu}{in}\mspace{14mu}(21)}:} & \; \\\begin{matrix}{\frac{\upsilon_{o}}{R_{2}} = {{{- v_{o}}\; R_{3}\; C_{2}\;\left( {s - {j\;\omega_{c}}} \right)\;\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)} -}} \\{i_{x}\; R_{3}\;\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)}\end{matrix} & (24) \\{{\upsilon_{o} = \frac{{- i_{x}}\; R_{3}\;\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)}{{R_{3}\; C_{2}\;\left( {2 - {j\;{\omega\;}_{c}}} \right)\;\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)} + \frac{1}{R_{2}}}}{{{Equating}\mspace{14mu}(20)\mspace{14mu}{and}\mspace{14mu}(25)}:}} & (25) \\{i_{x} = \frac{- i_{1}}{R_{3}\;\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + {sC}_{1}} \right)}} & (26) \\{\mspace{20mu}{= {\frac{- i_{1}}{R_{3}\; C_{1}}\;\left( \frac{1}{s + \frac{\omega_{0}}{Q}} \right)}}} & (27)\end{matrix}$

Using (27), (16), and (15): $\begin{matrix}{i_{x} = {{- \omega_{c}}\;{C_{1}\left\lbrack {{- R_{3}}\; C_{2}\;\left( {s - {j\;\omega_{c}}} \right)} \right\rbrack}\frac{- 1}{R_{3}\; C_{1}\;\left( {s + \frac{\omega_{0}}{Q}} \right)}\;\upsilon_{o}^{*}}} & (28) \\{= {{- \frac{\omega_{c}\; C_{2}\;\left( {s - {j\;\omega_{c}}} \right)}{s + \frac{\omega_{0}}{Q}}}\;\upsilon_{o}^{*}}} & (29) \\{= {{{- \frac{\omega_{c}\;{sC}_{2}}{s + \frac{\omega_{0}}{Q}}}\;\upsilon_{o}^{*}}\; - {\frac{\omega_{c}^{2}\; C_{2}}{s + \frac{\omega_{0}}{Q}}\;\upsilon_{o}\mspace{14mu}\left( {{{Note}:\mspace{14mu}{j\;\upsilon_{o}^{*}}} = {- \upsilon_{o}}} \right)}}} & (30)\end{matrix}$

Each term in (30) describes an admittance function. The first term hasthe form: $\begin{matrix}{i = {K\;\frac{s}{s + p}\;\upsilon}} & (31)\end{matrix}$

Consider the passive network in FIG. 7. $\begin{matrix}{i = {\frac{1}{R_{x}}\;\frac{s}{s + \frac{1}{R_{x}\; C_{x}}}\;\upsilon}} & (32)\end{matrix}$

Comparing terms: $\begin{matrix}{R_{x} = \frac{1}{\omega_{c}\; C_{2}}} & (33) \\{C_{x} = \frac{1}{R_{x}\;\frac{\omega_{0}}{Q}}} & (34) \\{\mspace{14mu}{= \frac{\omega_{c}\; C_{2}}{\frac{\omega_{0}}{Q}}}} & (35)\end{matrix}$

The second term in (30) has the form: $\begin{matrix}{i = {K\;\frac{1}{s + p}\;\upsilon}} & (36)\end{matrix}$

The natural choice is an R-L series network as in FIG. 8 $\begin{matrix}{i = {\frac{1}{R_{y} + {sL}_{y}}\;\upsilon}} & (37) \\{= {\frac{1}{L_{y}}\;\frac{1}{s + \frac{R_{y}}{L_{y}}}}} & (38)\end{matrix}$

Although the form of the transfer function is correct, the inductor isnot suitable for IC implementation.

Consider the “T” network in FIG. 9 $\begin{matrix}{i = {\frac{1}{R_{y}^{2}C_{y}}\frac{1}{s + \frac{2}{R_{y}C_{y}}}\upsilon}} & (39)\end{matrix}$

Comparing terms: $\begin{matrix}{R_{y} = {\frac{1}{2\;\omega_{c}^{2}\; C_{2}}\;\frac{\omega_{0}}{Q}}} & (40) \\{C_{y} = {\frac{2}{R_{y}}\;\frac{1}{\frac{\omega_{0}}{Q}}}} & (41) \\{\mspace{79mu}{= {4\;\omega_{c}^{2}\; C_{2}\;\left( \frac{\omega_{0}}{Q} \right)^{2}}}} & (42)\end{matrix}$

FIG. 10 is a block diagram illustrating two second order filters (onefor each channel) each implemented using a single op amp. ResistorR_(z)=1/ω_(c)C₂ implements the controlled current source i₂. Secondorder filter 1002 is implemented in the I channel and Second orderfilter 1012 is implemented in the Q channel. In the embodiment shown,inverting buffers 1003 and 1013 provide inverted outputs. In a fullydifferential implementation, both polarities of the output areavailable; therefore, the inverting buffers are not needed. Passivenetwork 1004 enables an output IO from the I channel to be fed intoinput Q2 of the Q channel. Likewise, passive network 1014 enables anoutput QO from the Q channel to be fed into an input I2 of the Ichannel. It should be noted that the coupling network shown in FIG. 10is exemplary and in other embodiments, other coupling networks are used.

A complex multiple feedback filter has been described. A multiplefeedback filter that includes two poles and a single op amp isimplemented in the I and Q channels. A first linking network ofcapacitors and resistors links a Q channel output back to an I channelinput and a second linking network of capacitors and resistors links anI channel output back to a Q channel input. In this manner, a complexfilter is implemented with two poles and only one op amp in the Ichannel and one op amp in the Q channel.

Although the foregoing invention has been described in some detail forpurposes of clarity of understanding, it will be apparent that certainchanges and modifications may be practiced within the scope of theappended claims. It should be noted that there are many alternative waysof implementing both the process and apparatus of the present invention.Accordingly, the present embodiments are to be considered asillustrative and not restrictive, and the invention is not to be limitedto the details given herein, but may be modified within the scope andequivalents of the appended claims.

1. A complex filter comprising: an I channel having a first I channeloutput and a second I channel output; a Q channel having a first Qchannel output and a second Q channel output; wherein the second Ichannel output is input to the Q channel through a first passive networkand wherein the second Q channel output is input to the I channelthrough a second passive network, and the first passive network and thesecond passive network are RC networks.
 2. A complex filter as recitedin claim 1 wherein the second Q channel output is a low impedanceoutput.
 3. A complex filter as recited in claim 1 wherein the second Ichannel output is a low impedance output.
 4. A complex filter as recitedin claim 1 wherein the second Q channel output is at the output of an opamp.
 5. A complex filter as recited in claim 1 wherein the second Ichannel output is at the output of an op amp.
 6. A complex filter asrecited in claim 1 wherein the second I channel output is input to the Qchannel at an input of an op amp.
 7. A complex filter as recited inclaim 1 wherein the second Q channel output is input to the I channel atan input of an op amp.
 8. A complex filter as recited in claim 1 whereinthe complex filter is used for image rejection.